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psychology of abstraction Post Date: Sun, 10 Aug 2008 19:42:25 +0000 We shall therefore close the special treatment of geometry and turn for a time to the analysis of algebra, returning, after some discussion of algebra, to the question of the relation between geometry and the other branches of mathematics. The psychological analysis of algebra must be introduced by a consideration, first, of the psychology of number, and second, by a consideration of the. psychology of abstraction. Autor of the post: Undefined The mental process which appears Post Date: Sun, 10 Aug 2008 19:29:25 +0000 Algebra is a generalized arithmetic, and as such is related historically to the science of number rather than to the science of space. This historical relation was the subject of brief comment in an earlier chapter and need not be elaborated here. The mental process which appears in number conscious- ness is in essence one of the many forms of comparison. Autor of the post: Undefined The word " calculate " Post Date: Sun, 10 Aug 2008 19:16:37 +0000 The objects to be counted are related in consciousness to some set of objects, or to some standard series of ideas, which serve as tallies. Thus, when primitive man came to possess so many cattle or tents that he could no longer remember them individually, he began to compare his pos- sessions to convenient and more readily manageable sets of objects which served as tallies. The word " calculate " and related words show that the ancient Roman used peb- bles (calculus, " a pebble ") for the purpose in question. Autor of the post: Undefined That is, if a pebble Post Date: Sun, 10 Aug 2008 19:05:57 +0000 He looked at one sheep or one ox and set aside a pebble as a reminder of this item of his wealth. The pebble thus set aside was easy to manage, and it was a safe reminder, for it was soon found to be invariably true that if there had once been a one-to-one correspondence between tallies and objects, there would always be a one-to-one correspondence. That is, if a pebble had been set aside for a sheep, then later there must be a sheep for each pebble. Autor of the post: Undefined We can understand this influence Post Date: Sun, 10 Aug 2008 18:50:25 +0000 The common series on which primitive man depended to thus remind himself of his possessions was the series of digits which he always had as convenient objects of im- mediate reference. The fingers exercised a significant and formative influence on the methods of counting. We can understand this influence only when we consider many facts regarding the fingers as a system of tallies. Autor of the post: Undefined The number system becomes, therefore Post Date: Sun, 10 Aug 2008 18:32:07 +0000 Thus we may comment in detail on one essential characteristic of the fingers which influenced the mind in making up number systems. The fingers are a limited group and require, in addition to themselves, certain supplementary devices which make it possible to use them over and over again in tally- ing off large collections of objects. The number system becomes, therefore, a series of multiples, and much of the terminology of number deals with this repeating of the base. Autor of the post: Undefined In the se- lection Post Date: Sun, 10 Aug 2008 18:13:24 +0000 Furthermore, the decimal base and the base of five which ap- pear in many number schemes are not mathematically nec- essary. The base might equally well have been six or twelve or some other small, easily recognized number. In the se- lection of ten we see the influence of the fingers. Autor of the post: Undefined The number terms used Post Date: Sun, 10 Aug 2008 18:01:50 +0000 In the struggles through which primitive people passed in develop- ing a number terminology which should include many tens, we see the embarrassments and limitations which grew out of the limited system of tallies which the hand made available. After primitive man became familiar with the advantages of a one-to-one comparison of objects with a system of tal- lies, it was natural that he should translate the system of tallying which he had evolved into a system of words. The number terms used by primitive peoples exhibit almost universally the direct relation of number words to the hand- tally system. Autor of the post: Undefined The creation of a series Post Date: Sun, 10 Aug 2008 17:42:43 +0000 Thus one historian 1 remarks: The number five is generally represented by the open hand, and it is said that in almost all languages the words " five " and " hand " are derived from the same root. The only tribes of whom I have read who did not count in terms either of five or of some multiple of five are the Bolans of West Africa, who are said to have counted by multiples of seven, and the Maoris, who are said to have counted by multiples of eleven. The creation of a series of words which could be substi- tuted for the objects used as tallies is a long process. Autor of the post: Undefined Evi- dently such a series Post Date: Sun, 10 Aug 2008 17:26:55 +0000 Such a series of words has the disadvantage over a series of tal- lies that the word disappears as soon as it is uttered. Imag- ine a speaker saying over to himself, " one, one, one," and so on, as the items of his property pass in review. Evi- dently such a series of ones is of no use in keeping tally, because the tally system is less easy to remember than the objects taken directly. 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